>> Is that what you're talking about? Rhetoric?>> No, not rhetoric. Even though some rhetoric is based on formal reasoning, formal reasoning is not a required element of rhetoric, persuasion is. Formal reasoning is the ability to understand and work with an abstract theory. It is pedantic (for lack of a better word). Persuasion is not an element of formal reasoning.>> Note: All theories are abstract so "abstract theory" is redundant.>

So I'm still wondering if your touristic excursions into math teachingpedantry is "formal reasoning". Apparently you're not equating itwith "formal logic" ala the logic of Frege - Russell - Wittgenstein(the latter in fragments, as by Philosophical Investigations it'smostly prose, though deeply worked to be grammatical in a certainway).

>>> I am good with the extra for experts inserts, but you have to get the>>> algebra first before you can teach the student the why behind it, otherwise

I think saying you're not needing to give "the why behind it" up frontis what too many teachers are saying: we'll tell you later what thisis for, "just trust us" or "you need to know it because it's on thetest" (a smug tautology -- don't let your teacher get away with suchcheap and easy retorts, have some standards!).

There's a credibility problem among math teachers that's mostlyovercome in STEM, which supplies its own context. Add physics tocalculus and voila, a healthy brew. You can subtract the physics awayagain, but at least you knew enough to stick with hybrids. STEM isall about hybridizing the disciplines, not keeping them "pure" (thatfixation of "purity" was a horrible mistake that so many fell into,bless their souls).

Speaking of which: a shout out to Dan Suttin of OCTA-TETRA Museum, inSan Antonio, TX:

>>> it is just pretend. Besides, Clyde will tell you that since they are doing>>> algebra, and since algebra is controlled by group theory, they understand>>> group theory. So, no reason to teach group theory twice.:)>>>>>>> This is reinforcing the stereotype that you need to be especially>> gifted to get it in a different sequence.>> No, I am only saying that you have to get algebra before you attempt to get the theory of algebra. In the context of my statement, expert simply means that the kid got algebra fairly well.>

That's very confusing to students, to say we can't give you theorybefore we give you practice. Why not both together? What's so hardabout Closure, for example?

We're just saying if you have two of this type of something, like aturtle or integer or donut or fruit, that if you combine them using anoperation we'll call "multiply" (why not), that you always get aresulting output of the same type.

Two donuts make a donut; two fruits make a fruit. But then thesearen't intuitive examples. What's intuitive is "begetting".

My hypothesis is the reason we don't get into group theory that muchwith kids has more to do with the schoolish bias against R-ratedtopics, i.e. sex.

Fibonacci numbers had to do with rabbits mating, but how many textbooks for pre-17 year olds dare mentioning such a thing? Censoring inthis way is a neo-Victorian reflex.

In the meantime kids are "trash talking" at recess and on the bus inso many R- and X-rated ways, comparing household yammerings, gettingan ethnographic mix (good prep for later prez types). Welcome toSpike Lee movies and life in the big city.

Speaking in general terms of "types begetting types" is what BibleCamp might be about, since Genesis has lots of begets in it.

That's where the puritanical will tolerate maybe a tiny bit of sextalk: in a Biblical context. Heaven forbid though, that secularschools run by the gummint should ever talk about rabbits "doing it"(gummint sex education was the reason many would home school). NoFibonacci numbers until you're in college!

In other words, to use 60s talk: teachers are / were too "hung up" toteach higher math, because higher math is metaphoric and metaphors aredangerously day dreamy and promoting of imagination.

Staid arithmetic, on the other hand (like what Paul teaches), safelykills or neutralizes the imagination, sweeps it clean of all objectsbut an abacus (if you're Japanese), or a calculator if you're American(just press the buttons, what's with all this finger twitching?).

>>> The segment I'm talking about involves distilling the totatives of a>> number N, using the GCD algorithm and then showing how totatives>> multiplied modulo N have>>>> (a) Closure>> (b) Associativity>> (c) Inverse elements>> (d) an Neutral element>>>> (CAIN -- plus this group is also Abelian (Biblical pun)).>>>> Explain what each of those properties of a group means. Play around>> with more of them. It's simple stuff, easy peasy. Amenable to>> "gamefication".>>

Inverse and Neutral go together.

Every rabbit has a perfectly opposite rabbit such that, were these twoto find each other and have a child (unlikely), that child would be aNeutral One.

The Neutral One, when multiplied with another rabbit, gives a clone ofthat other rabbit. A miracle. Like a black swan (these do exist bythe way -- some philosophers pretend otherwise).

In the integer world (domain) we call the rabbits "ints" and

((3 * 1) == 3) is True

(that's Python syntax), just as

((1 * 4) == 4) is True

i.e. each "coupling" with a Neuter gives a Same (clone). Lots ofgames we could play, spiraling into it.

>> This is what's called "spiraling" by the way, where you go into>> something a little, from one angle, and then get into it more later,>> from another angle. John Saxon was emphatic about "spiraling".>> Yeah, well that isn't spiraling. Spiraling starts with a foundational treatment and the "spiraling" occurs after that, not before. What you wrote would be called enrichment or extra for experts.>

There's nothing more foundational than the Bible and Genesis accordingto many Americans, so that might be how we'll start at those summercamps (when we're away from those public schools and their secularways).

Another beget story goes like this (the operation, instead of"multiply" is called "dualing" -- not "dueling"), a unary operation:

Combine is a binary operation, a function of two arguments, or,curried: Combine(Tetrahedron) is a function in search ofDual(Tetrahedron) as its argument. This feeds in to Haskell and, moregenerally (when we do some more turns) Lambda Calculus.

In Haskell, Tetrahedron `Combined` (Dual Tetrahedron) would be how touse 'infix' notation instead.

That's probably what you mean by the difference between Algebra andAlgebra Theory, like the difference between Haskell and LambdaCalculus?

In that sense, there are many algebras.

Anyway, none of the above is difficult and if that's the curriculum inSan Antonio, but not Sarasota, then so what?

Not everyone needs to be on the same page, in this great country of ours.

So lets *never* pretend we're in search of the one great NationalCurriculum that all must adhere too.

That'd be anti-American both in spirit and in practice. I would neverwant to be associated with such an unpatriotic endeavor.

>>>> Junior is saddled with the same>>> linear sequence is grandparents had. Is that a good thing? By>>> definition?>>>>>>>>> Well, Junior is just human, like his grandparents. Naturally, the>>> progression would be the same, right?>>>>>>> Is this what you call "formal reasoning" then?>>>> You seem to consider yourself a good example of what your favored>> curriculum would turn out.>> I would like more students to think rationally.>>>>>> I assume we're being treated to an example of what "reasoning" means,>> am I safe to assume that?>> Yes.>

Different from just passing the Turing Test I assume. One doesn'tneed to be especially rational to pass the Turing Test.

> While a forum like this would be classified as rhetoric, my rhetoric is based mostly on formal reasoning and yours mostly on ideology and word play.>

I teach logical thinking for a living one could say. To hundreds. Ithink my rhetoric is really top notch, one of the fastest horses onthe track (some days).

The points I've been making, that the curriculum is far from fixed,and that my New Math education was already different than yours, evenone generation apart... did those points register at all? Do youadmit that the sequence is far from fixed?

> I have this theory that there is a natural progression to all of this. When we lay down the ideas and theory of mathematics in a student, that process inevitably follows the same pattern in which it was discovered by humankind.

>I am not saying that we must visit every success and failure of the past, but the progression of sophistication is the same. This isn't a unique theory, it is known fairly well in art and music. When you study art through history, the topics like form, shadow, foreground, perspective, appear in the same order as they do when we teach art. This is because children begin with the same primitive understanding of art as did primitive adults in the past. And they advance through the layers of sophistication in art in the same way that the generations of artists did, in the past. Whether it is one child learning art, or the whole human race discovering it, the progression is the same. It is quicker for the modern child (it doesn't take 100's or 1000's of years) because they do not have to discover it as did humankind, we are able to teach it to them. Otherwise we would stay forever at square one. Irregardless of the fact that we can teach, we are still bound by the same natural p!rogression.>

These types of theories usually assume the present civilization is apinnacle i.e. it was all leading up to now.

Stephan J. Gould and Lynn Margulis on the other hand, were more intoseeing "evolution" as "transformation" i.e. it meanders, wanders,adapts. It's not getting "more this" or "more that" overall i.e. it'snot a progression so much as a random exploration of a possibilityspace, a continuous morphing.

Humanity is not a pinnacle of anything cosmically speaking and ourcurrent civilization is not a pinnacle of anything either. Westumbled into this niche and we'll stumble out of it. So it goes.

> When a child begins the journey of acquiring the theory of mathematics they are very much like our settlers on the new earth starting from scratch. In fact, as the child advances in mathematics aren't they advancing the state of the art of their own thinking? They are building a sort of technology, a technology of mind. Just like are settlers, they have to go through the progression and just like our settlers they have to linger a bit at each iteration to ensure that the infrastructure they are building will support the next iteration.>

"The theory of mathematics" -- as if we all knew what that means, oras if there were only one. Is that what's believed in the state ofFlorida?

> The settlers already had the blueprint for technology. They knew the journey. When teaching a chid however, it is the teacher that holds that information. They have been there and done that. It is their job to guide the child on that journey.>> Bob Hansen

Lets not forget the Library phase of life.

In many cultures, it's important for neophytes / noobs to wander insearch of a teacher or teachers. It's not like you just immediatelyhave the right teachers at hand, just because that's how they wereassigned by the union and/or district on the basis of seniority orwhatever politics. No, you must search in the Library.

In the old days, that might mean leaving Florida, as Florida was notknown to have wise people or great libraries, but only those seekingthe Fountain of Youth (i.e. misguided cretins over-due for rebirth).

Today, however, there's Youtube, and you can seek teachers from almostanywhere. A man named Khan has been popular. A young woman named Vihas a following (they've collaborated as each recognizes the other astalented). Find a spelling teacher while you're at it.

Find people who speak your language, relate to your concerns.

In this model, the job of early teachers is to teach reading andresearch skills, some alpha-numeracy, and then to encourage the use ofthese skills to browse the Internet in search of their next teachers.

They find some locally as well.

Parents are a good start, but they're often not home in primitivesocieties that aren't smart enough to arrange for parenting by parents(the USA had fallen to this extreme low of cultural IQ in many zipcodes by the late 1900s, before telecommuting and village livingreally got going again).

The Dual-Combine way of introducing Polyhedrons, based on the model ofGenesis, may lead to more R-rated segments than some schools allow.

Fortunately, as an andragog (a teacher of adults), I'm not obligatedto censor my language as much as the pedagogs.

The Friends (religious group) have also come to realize that their"plain speech" testimony is consistent with R rated speech, i.e. themore explicit vernacular of the playground and Hollywood films issometimes the fastest way to get into the mathematics at hand.